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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习2.2-2.3}
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\date{2024 年 3 月 25 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
\begin{enumerate}[label={(\arabic*)}]
\item  写出指数函数的定义。
\item  写出正弦函数、余弦函数、正切函数和余切函数的定义。
\item  写出双曲正弦函数、双曲余弦函数、双曲正切函数和双曲余切函数的定义。
\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 02
\begin{enumerate}[label={(\arabic*)}]
\item  计算 $\sin(1+2i)$ 的值。
\item  解释极限 $\lim\limits_{z\to\infty} e^z$ 不存在。
\item  解释 $|\sin z|\le 1$ 和 $|\cos z|\le 1$ 不一定成立。
\item  证明 $\sin^2z + \cos^2z=1$ 仍然成立。
\end{enumerate} 


\vspace{0.2cm}

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\item  %Problem 03
\begin{enumerate}[label={(\arabic*)}]
\item  写出幅角函数 $w=\mathrm{Arg}\,(z)$ 的定义。
\item  写出根式函数 $w=\sqrt[n]{z}$ 的定义。
\item  写出对数函数 $w=\mathrm{Ln}\,(z)$ 的定义。
\item  写出一般幂函数 $w=z^\alpha$ 与一般指数函数 $w=\alpha^z$ 的定义。
\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 04
设 $f(z)=\sqrt[3]{z}$ 确定在 $\mathbb{C}-(-\infty,0]$ 上，并且 $f(i)=-i$, 试求 $f(-i)$ 的值。

\vspace{0.2cm}

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\item  %Problem 05
\begin{enumerate}[label={(\arabic*)}]
\item  计算 $\mathrm{Ln}\, (i)$.
\item  计算 $\mathrm{Ln}\, (3+4i)$.
\item  计算 $i^i$
\item  计算 $2^{1+i}$.
\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 06
求函数 $f(z) = \sqrt{z(1-z)}$ 的支点。

\vspace{0.2cm}

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\item  %Problem 07
证明 $f(z)=\sqrt[3]{z(1-z)}$ 在将 $z$ 平面适当割开后，能分出三个单值解析分支。并求出在 $z=2$ 取负值的那个分支在 $z=i$ 的值。

\vspace{0.2cm}

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\item  %Problem 08
证明函数 $f(z)=\mathrm{Ln}\,(1-z^2)$ 在割去从 $-1$ 到 $i$ 的直线段和从 $i$ 到1的直线段、与射线 $x=0$且 $y\ge 1$ 的 $z$ 平面内能分出单值解析分支。并求 $z=0$ 时函数值等于零的那一支在 $z=2$ 的值。

\vspace{0.2cm}

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\item  %Problem 09
写出反正切函数、反正弦函数、反余弦函数的定义。用对数函数表示这些反三角函数。

\vspace{0.2cm}

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\item  %Problem 10
\begin{enumerate}[label={(\arabic*)}]
\item  计算 $\mathrm{Arcsin}\, (2)$.
\item  计算 $\mathrm{Arctan}\, (2i)$.
\end{enumerate} 

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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